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Chapter 4: Q.4.61 (page 167)
The probability of being dealt a full house in a hand of poker is approximately . Find an approximation for the probability that in hands of poker, you will be dealt at least full houses.
An approximation for the probability that in hands of poker, we will be dealt at least full houses is .
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Three dice are rolled. By assuming that each of the possible outcomes is equally likely, find the probabilities attached to the possible values that X can take on, where X is the sum of the 3 dice.
When coin 1 is flipped, it lands on heads with probability .4; when coin 2 is flipped, it lands on heads with probability .7. One of these coins is randomly chosen and flipped 10 times.
(a) What is the probability that the coin lands on heads on exactly 7 of the 10 flips?
(b) Given that the first of these 10 flips lands heads, what is the conditional probability that exactly 7 of the 10 flips land on heads?
Suppose that a biased coin that lands on heads with probability is flipped times. Given that a total of heads results, find the conditional probability that the first outcomes are
(a) (meaning that the first flip results in heads, the second is tails, and the third in tails);
(b)
Here is another way to obtain a set of recursive equations for determining , the probability that there is a string of consecutive heads in a sequence of flips of a fair coin that comes up heads with probability :
(a) Argue that for , there will be a string of consecutive heads if either
1. there is a string of consecutive heads within the first flips, or
2. there is no string of consecutive heads within the first flips, flip is a tail, and flips are all heads.
(b) Using the preceding, relate . Starting with , the recursion can be used to obtain , then, and so on, up to .
There are two possible causes for a breakdown of a machine. To check the first possibility would cost C1 dollars, and, if that were the cause of the breakdown, the trouble could be repaired at a cost of R1 dollars. Similarly, there are costs C2 and R2 associated with the second possibility. Let p and 1 鈭� p denote, respectively, the probabilities that the breakdown is caused by the first and second possibilities. Under what conditions on p, Ci, Ri, i = 1, 2, should we check the first possible cause of breakdown and then the second, as opposed to reversing the checking order, so as to minimize the expected cost involved in returning the machine to working order?
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